Consider the Hecke algebra $H_n$ of type $A_{n-1}$ with standard basis $T_w$, $w \in S_n$ with the quadratic relations $(T_s - u) (T_s + u^{-1}) = 0$ and braid relations. The unsigned canonical basis $C'_w$, $w \in S_n$ gives rise to a basis for the irreducible $H_n$-module $M_\lambda$ of shape $\lambda$: fix some SYT $T$ of shape $\lambda$; then this basis is {$C'_w : P(w) = T$} after quotienting by cells lower down in the Kazhdan-Lusztig preorder and $P(w)$ is the insertion tableau of $w$.

Similarly, $M_\lambda$ has a basis coming from the signed canonical basis $C_w$, $w \in S_n$.

What is known about the transition matrix between these two bases? Does it become the identity matrix at $u=0$?

This seems to be trickier to understand than the transition matrix between all the $C$s and all the $C'$s. I expect I will be able to prove the second question on my own, but I'd rather cite it if it's in the literature somewhere.

2 Answers
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Not much seems to be known about this matrix in general, and it does become the identity at $u=0$. I show this in the paper Quantum Schur-Weyl duality and projected canonical baseshttp://arxiv.org/abs/1102.1453
using quantum Schur-Weyl duality and its compatibility with canonical bases. This only proves it in type A. I do not know what happens in other types.